Properties

Label 2-2001-1.1-c1-0-19
Degree $2$
Conductor $2001$
Sign $1$
Analytic cond. $15.9780$
Root an. cond. $3.99725$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 4·5-s − 4·7-s + 9-s + 4·11-s + 2·12-s − 5·13-s − 4·15-s + 4·16-s − 5·17-s + 5·19-s − 8·20-s + 4·21-s + 23-s + 11·25-s − 27-s + 8·28-s − 29-s − 2·31-s − 4·33-s − 16·35-s − 2·36-s + 5·37-s + 5·39-s − 2·41-s + 43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 1.78·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 1.38·13-s − 1.03·15-s + 16-s − 1.21·17-s + 1.14·19-s − 1.78·20-s + 0.872·21-s + 0.208·23-s + 11/5·25-s − 0.192·27-s + 1.51·28-s − 0.185·29-s − 0.359·31-s − 0.696·33-s − 2.70·35-s − 1/3·36-s + 0.821·37-s + 0.800·39-s − 0.312·41-s + 0.152·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(15.9780\)
Root analytic conductor: \(3.99725\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2001} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.223221246\)
\(L(\frac12)\) \(\approx\) \(1.223221246\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.260867415265607682796848286962, −8.964560335588406953513666380604, −7.28994226615198199606122345411, −6.60123660484370742188522259092, −5.96397978578636854059591521654, −5.27029940034070622731192453076, −4.43152301254281187157853772822, −3.28897286527146816079042547184, −2.16848873486685433534249006869, −0.76225775874813821996698126597, 0.76225775874813821996698126597, 2.16848873486685433534249006869, 3.28897286527146816079042547184, 4.43152301254281187157853772822, 5.27029940034070622731192453076, 5.96397978578636854059591521654, 6.60123660484370742188522259092, 7.28994226615198199606122345411, 8.964560335588406953513666380604, 9.260867415265607682796848286962

Graph of the $Z$-function along the critical line